Pol

$$x$$

A formal indeterminate (formal variable) which may be used to construct polynomials or rational functions as algebraic objects.

$$x_{n}$$

A formal indeterminate (formal variable) which may be used to construct polynomials or rational functions as algebraic objects, distinguished by an index $n$.

$$\left(x_{1}, x_{2}, x_{3}\right)$$

Three formal indeterminates.

$$x_{1} \ne 0 \;\mathbin{\operatorname{and}}\; x_{1} \ne x_{2}$$

A formal indeterminate is not the same thing as an expression-level variable; it is a concrete mathematical object, and in particular an object distinct from any number and from any other indeterminate.

$$x_{n} - x_{n} = 0 x_{n} = 0 \;\mathbin{\operatorname{and}}\; {x}_{n}^{0} = \frac{x_{n}}{x_{n}} = 1$$

Indeterminates of this type generates algebras with ordinary numbers as coefficients. They are not suitable for working in rings with different Zero and One elements.

$$x_{1} x_{2} = x_{2} x_{1}$$

Indeterminates of this type commutes with numbers and with other indeterminates of the same kind.

$${\left(\sum_{i=1}^{m} x_{i}\right)}^{n} = \sum_{k \in S} \frac{n !}{\prod_{i=1}^{m} \left({k}_{i}\right)!} \prod_{i=1}^{m} {x}_{i}^{{k}_{i}}\; \text{ where } S = \left\{ k : k \in {\{0, 1, \ldots, n\}}^{m} \,\mathbin{\operatorname{and}}\, \sum_{i=1}^{m} {k}_{i} = n \right\}$$

The multinomial theorem for formal indeterminates.

Last updated: 2020-03-06 00:22:16